Poisson: Create a Poisson distribution

In distributions3: Probability Distributions as S3 Objects

Create a Poisson distribution

Description

Poisson distributions are frequently used to model counts.

Usage

Poisson(lambda)

Arguments

lambda

The shape parameter, which is also the mean and the variance of the distribution. Can be any positive number.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail.

In the following, let X be a Poisson random variable with parameter lambda = \lambda.

Support: \{0, 1, 2, 3, ...\}

Mean: \lambda

Variance: \lambda

Probability mass function (p.m.f):

P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}

Cumulative distribution function (c.d.f):

P(X \le k) = e^{-\lambda} \sum_{i = 0}^{\lfloor k \rfloor} \frac{\lambda^i}{i!}

Moment generating function (m.g.f):

E(e^{tX}) = e^{\lambda (e^t - 1)}

Value

A Poisson object.

See Also

Other discrete distributions: Bernoulli(), Binomial(), Categorical(), Geometric(), HurdleNegativeBinomial(), HurdlePoisson(), HyperGeometric(), Multinomial(), NegativeBinomial(), PoissonBinomial(), ZINegativeBinomial(), ZIPoisson(), ZTNegativeBinomial(), ZTPoisson()

Examples

set.seed(27)

X <- Poisson(2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

distributions3 documentation built on Sept. 30, 2024, 9:37 a.m.